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PMATH 945

Topics in Algebra

Graduate

Winter 2027 · Sheaves and Homological Algebra

Topics

  • Homological algebra
  • Categories of sheaves
  • Derived categories and applications
Winter 2026 · Category Theory and Homological Algebra

Topics

  • The topics covered in the course are:
  • Categories and functors
  • Natural transformations
  • The opposite category
  • Adjoints
  • Tensor-Hom adjunction
  • Yoneda’s lemma
  • Initial and terminal objects
  • Limits and colimits
  • Abelian categories
  • Mitchell’s embedding theorem
  • Projective modules and vector bundles
  • Injective modules
  • Complexes, long exact sequences, homotopies of complexes 1
  • Projective resolutions
  • Derived functors
  • Ext and Tor
  • Ext via Yoneda equivalence Some topics may be omitted and others may be added depending on time. Assessments
  • Submit your solutions as PDFs via Crowdmark.
Fall 2026 · Arithmetic Dynamics

Topics

  • Dynamical Mordell–Lang problem
  • Linear recurrence sequences
  • Polynomial–exponential Diophantine equations
  • Algebraic geometry and valuations
  • Skolem–Mahler–Lech theorem and étale maps
  • Noetherian topological spaces
  • Automata and Derksen’s positive-characteristic analogue

Readings

  • Instructor course notes to be distributed.
    unspecified
Winter 2025 · Topics in Algebra - Modern Algebraic Geometry

Topics

  • Schemes and sheaves
  • Affine and projective schemes
  • Morphisms and fibre products
  • Dimension and regularity
  • Line bundles
  • Riemann–Roch and Serre duality for curves
Fall 2024 · Rings and their applications

Topics

  • Jacobson density theorem and radical
  • Artin–Wedderburn theorem
  • Goldie’s theorem and noncommutative localization
  • Polynomial-identity theory
  • Gelfand–Kirillov dimension
  • Golod–Shafarevich theorem
  • Brauer groups and cohomology
Fall 2021 · Category O: A Second Course in Lie Algebras

Topics

  • Structure theory of Lie algebras
  • Category O and infinite-dimensional representations
  • Verma modules
  • Weyl character formula
  • Translation functors
  • Possible flag-variety localization and Cherednik-algebra generalizations
Winter 2020 · Applications of Operator Algebras in QIT

Topics

  • C*-algebras, operator systems and operator spaces in quantum information
  • Completely positive and completely bounded maps
  • Compact operators and Schatten classes
  • von Neumann algebras and GNS theory
  • Stinespring dilation
  • Knill–Laflamme theory, private channels and quantum marginals
  • Capacity theory and Bell-inequality violations
Fall 2020 · Clifford Algebras, Spinors, and Calibrations

Readings

  • Chevalley, The algebraic theory of spinors , Columbia University Press, 1954.
    unspecified
  • Conway and Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry , A K Peters, 2003.
    unspecified
  • Harvey, Spinors and calibrations (Perspectives in Mathematics), Academic Press, 1990.
    unspecified
  • Lawson and Michelson, Spin geometry (Princeton Mathematical Series), Princeton University Press, 1989.
    unspecified
  • Porteous, Clifford algebras and the classical groups (Cambridge Studies in Advanced Mathematics), Cam- bridge University Press, 1995.
    unspecified
  • There is no required textbook for
    unspecified
Winter 2019 · Algebraic Constructions

Topics

  • Generators and relations
  • Free groups, free algebras and rewriting systems
  • Gröbner–Shirshov bases
  • Burnside problems
  • Golod–Shafarevich groups
  • Growth and Gelfand–Kirillov dimension
  • Division and enveloping algebras
  • Poincaré–Birkhoff–Witt theorem
  • Tensor products and tensor categories
  • Ultraproducts and Malcev’s theorem

Readings

  • Instructor course notes to be distributed.
    unspecified
Fall 2019 · Approximate Representation Theory of Groups and Non-local Games

Topics

  • The course will cover some of the basics of approximate representation theory, focusing on stability, finite-dimensional approximations, and applications to nonlocal games and ran- domized benchmarking.

Readings

  • Approximate representation theory is a new subject, so there is no textbook we can use. I will post lecture notes to the course website, as well as a list of potential references.
    unspecified
Fall 2019 · Category Theory and Homological Algebras

Topics

  • Categories and functors, natural transformations, the opposite category, adjoints, tensor-Hom adjunction, Yoneda’s lemma, initial and terminal objects, limits and colimits, Govorov-Lazard theorem and filtered subcategories, Abelian categories, Mitchell’s embedding theorem, Projective modules and vector bundles, injective modules, complexes, long exact sequences, homotopis of complexes, projective resolutions, derived functors, Ext and Tor, Ext via Yondeda equivalence.
Sources